3.1 Hierarchy of vortices

Figure 3.

Isosurfaces of the second invariant $Q^{+}~(=1.6\times 10^{-3})$ of the velocity gradient tensor. The threshold is set as $m_{Q}+0.4s_{Q}$ . The colour indicates the streamwise velocity $u$ , ranging from $u=0$ (blue) to $u=U_{\infty }$ (red). Half of the spanwise direction of the computational domain is visualized. The Reynolds number $Re_{\unicode[STIX]{x1D703}}$ changes from $2940$ to $3170$ in the streamwise direction. The flow is from lower left to upper right. The black grid width is $200$ wall units evaluated at the plane corresponding to $Re_{\unicode[STIX]{x1D703}}=2860$ , and the blue solid line indicates the $99\,\%$ boundary layer thickness  $\unicode[STIX]{x1D6FF}_{99}$ .

To capture vortical structures in turbulent shear flows, the second invariant $Q~(=1/2\unicode[STIX]{x1D6FA}_{ij}\unicode[STIX]{x1D6FA}_{ij}-1/2\unicode[STIX]{x1D61A}_{ij}\unicode[STIX]{x1D61A}_{ij})$ of the velocity gradient tensor is widely used, where $\unicode[STIX]{x1D6FA}_{ij}$ and $\unicode[STIX]{x1D61A}_{ij}$ are the rate-of-rotation and rate-of-strain tensors. Setting the criterion as $Q>Q_{0}$ , with $Q_{0}$ being a positive threshold, we can identify vortical regions. Let us start by observing vortical structures identified by the $Q$ criterion without the coarse-graining. Previous DNS of turbulent boundary layers at high Reynolds numbers showed that hairpin vortices disappear in downstream regions (Jiménez et al. Reference Jiménez, Hoyas, Simens and Mizuno2010; Schlatter et al. Reference Schlatter, Li, Örlü, Hussain and Henningson2014). Our DNS also shows similar results. Figure 3 illustrates the vortical structures obtained by our DNS with a choice of $Q_{0}=m_{Q}+0.4s_{Q}$ , so that those in the log layer are visualized. Here, $m_{Q}$ and $s_{Q}$ are, respectively, the mean and standard deviation of $Q$ at $y^{+}=200$ and $Re_{\unicode[STIX]{x1D703}}=3170$ . Some of the vortices visualized in the region apart from the wall are arch-like, but they are not connected to streamwise vortices. Schlatter et al. (Reference Schlatter, Li, Örlü, Hussain and Henningson2014) also showed similar results by using the $\unicode[STIX]{x1D706}_{2}$ criterion (Jeong & Hussain Reference Jeong and Hussain1995). Here, $\unicode[STIX]{x1D706}_{2}$ is the second eigenvalue of $\unicode[STIX]{x1D61A}_{ik}\unicode[STIX]{x1D61A}_{kj}+\unicode[STIX]{x1D6FA}_{ik}\unicode[STIX]{x1D6FA}_{kj}$ . These DNS results imply that it is hard to identify hairpin vortices by using the $Q$ or $\unicode[STIX]{x1D706}_{2}$ criteria in a turbulent boundary layer at high Reynolds numbers. We must, however, emphasize that the velocity gradient tensor, and therefore $\unicode[STIX]{x1D6FA}_{ij}$ , $\unicode[STIX]{x1D61A}_{ij}$ , $Q$ and $\unicode[STIX]{x1D706}_{2}$ , are associated with the smallest-scale vortices. Therefore, the $Q$ or $\unicode[STIX]{x1D706}_{2}$ criterion identifies only the smallest-scale vortices. Multiscale vortices ranging from the length of the order of the boundary layer thickness to the Kolmogorov length coexist in the turbulence at high Reynolds numbers. Therefore, until we identify each scale of the vortices, we cannot approach the intrinsic vortex dynamics in a turbulent boundary layer.

Next, to identify such multiscale coherent structures, and to quantitatively verify the scenario proposed in § 1, we apply the Gaussian filter (2.1) to the velocity field. We show the isosurfaces of the second invariant $Q^{(\unicode[STIX]{x1D70E})}$ of the velocity gradient tensor coarse-grained at three length scales in figure 4, to see if a hierarchy of vortices can be captured by this coarse-graining method. The three filter scales are set as $\unicode[STIX]{x1D70E}^{\oplus }=10$ (figure 4 a), $40$ (figure 4 b) and $160$ (figure 4 c), where $\cdot ^{\oplus }$ denotes non-dimensionalization in wall units at $Re_{\unicode[STIX]{x1D703}}=2860$ . The arrows in figure 2(b) indicate the spanwise wavenumbers corresponding to these filter scales. We choose the thresholds of the isosurfaces as $Q_{0}=m_{Q}+0.5s_{Q}$ so that the vortices in the log layer are visualized. Figure 4 shows that a hierarchy of vortices indeed exists in the turbulent boundary layer. Noting that $200$ wall units are indicated by the grid width, we may see that the length scales of the vortices approximately correspond to the filter scales. In particular, we notice that at $\unicode[STIX]{x1D70E}^{\oplus }=160$ (figure 4 c) the larger-scale vortices have spatially organized structures and some of them are arch-like (see § 4.1 for more detailed discussions). We emphasize that these structures are invisible without the coarse-graining, even if we use lower thresholds of the isosurfaces. On the other hand, the small-scale vortices ( $\unicode[STIX]{x1D70E}^{\oplus }=10$ , figure 4 a) are similar to the structures (figure 3) without the coarse-graining.

Figure 4.

Isosurfaces of the second invariant of the velocity gradient tensor coarse-grained at (a) $\unicode[STIX]{x1D70E}^{\oplus }=10$ , (b) $40$ and (c) $160$ at the same instant and location as in figure 3. Thresholds are set as $Q=m_{Q}+0.5s_{Q}$ , namely, (a) $Q^{(\unicode[STIX]{x1D70E})+}=1.6\times 10^{-3}$ , (b)  $3.8\times 10^{-4}$ and (c)  $2.9\times 10^{-5}$ . The black grid width is $200$ wall units evaluated at the plane corresponding to $Re_{\unicode[STIX]{x1D703}}=2860$ , and the blue solid line indicates the $99\,\%$ boundary layer thickness $\unicode[STIX]{x1D6FF}_{99}$ . (d) All structures in (a–c) are simultaneously visualized.

3.2 Scale-dependent contributions to the stretching

The transport equation of the enstrophy $1/2|\unicode[STIX]{x1D714}_{i}|^{2}$ is expressed by

(3.1) $$\begin{eqnarray}\displaystyle \frac{1}{2}\frac{\text{D}|\unicode[STIX]{x1D714}_{i}|^{2}}{\text{D}t}=\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D61A}_{ij}\unicode[STIX]{x1D714}_{j}+\unicode[STIX]{x1D708}\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D714}_{i}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D714}_{i}$ is the vorticity. The enstrophy is amplified only when the enstrophy production term $\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D61A}_{ij}\unicode[STIX]{x1D714}_{j}$ is positively large, because the viscous term $\unicode[STIX]{x1D708}\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D714}_{i}$ weakens the enstrophy. We can therefore see how the vortices are amplified by investigating the magnitude of the enstrophy production term.

Figure 5.

Wall-normal distribution of each term of (3.5) at the streamwise location corresponding to $Re_{\unicode[STIX]{x1D703}}=3170$ . Red, $2\overline{\overline{\unicode[STIX]{x1D714}_{i}}\unicode[STIX]{x1D61A}_{ij}^{\prime }\unicode[STIX]{x1D714}_{j}^{\prime }}^{+}$ ; green, $\overline{\unicode[STIX]{x1D714}_{i}^{\prime }\overline{\unicode[STIX]{x1D61A}_{ij}}\unicode[STIX]{x1D714}_{j}^{\prime }}^{+}$ ; blue, $\overline{\unicode[STIX]{x1D714}_{i}^{\prime }\unicode[STIX]{x1D61A}_{ij}^{\prime }\unicode[STIX]{x1D714}_{j}^{\prime }}^{+}$ . The inset shows the close-up for the range $10^{2}\leqslant y^{+}\leqslant 10^{3}$ .

Figure 6.

Contribution (3.6) of the strain rate coarse-grained at $\unicode[STIX]{x1D70E}_{S}$ to the stretching of vortices at $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}=10~(\circ )$ , $20~(\Box )$ , $40~(\triangle )$ , $80~(\bullet )$ and $160~(\blacksquare )$ for $Re_{\unicode[STIX]{x1D703}}=3170$ at (a)  $y^{+}=300$ , (b)  $100$ and (c)  $40$ . The larger symbols correspond to the self-contribution ( $\unicode[STIX]{x1D70E}_{s}^{+}=\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}$ ). Blue symbols indicate the contributions (3.7) from the mean stretching.

For a turbulent boundary layer,

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{\unicode[STIX]{x1D714}_{1}}=\overline{\unicode[STIX]{x1D714}_{2}}=0, & \displaystyle\end{eqnarray}$$

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{\unicode[STIX]{x1D61A}_{13}}=\overline{\unicode[STIX]{x1D61A}_{31}}=\overline{\unicode[STIX]{x1D61A}_{23}}=\overline{\unicode[STIX]{x1D61A}_{32}}=\overline{\unicode[STIX]{x1D61A}_{33}}=0, & \displaystyle\end{eqnarray}$$

where

(3.4a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{i}=\overline{\unicode[STIX]{x1D714}_{i}}+\unicode[STIX]{x1D714}_{i}^{\prime },\quad \unicode[STIX]{x1D61A}_{ij}=\overline{\unicode[STIX]{x1D61A}_{ij}}+\unicode[STIX]{x1D61A}_{ij}^{\prime }.\end{eqnarray}$$

Here, the overbar $\overline{\,\cdot \,}$ denotes the average in the spanwise direction and time, the prime $\cdot \,^{\prime }$ denotes the deviation from the mean, and the subscripts $1$ , $2$ and $3$ denote the $x$ , $y$ and $z$ directions, respectively. Substituting (3.4) into the production term on the right-hand side of (3.1) and taking the average, we can rewrite it as

(3.5) $$\begin{eqnarray}\displaystyle \overline{\unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D61A}_{ij}\unicode[STIX]{x1D714}_{j}}=2\overline{\overline{\unicode[STIX]{x1D714}_{i}}\unicode[STIX]{x1D61A}_{ij}^{\prime }\unicode[STIX]{x1D714}_{j}^{\prime }}+\overline{\unicode[STIX]{x1D714}_{i}^{\prime }\overline{\unicode[STIX]{x1D61A}_{ij}}\unicode[STIX]{x1D714}_{j}^{\prime }}+\overline{\unicode[STIX]{x1D714}_{i}^{\prime }\unicode[STIX]{x1D61A}_{ij}^{\prime }\unicode[STIX]{x1D714}_{j}^{\prime }}. & & \displaystyle\end{eqnarray}$$

Here, we have used (3.2) and (3.3).

In figure 5, we show the wall-normal distribution of each term of (3.5) at the streamwise location corresponding to $Re_{\unicode[STIX]{x1D703}}=3170$ . The first term $2\overline{\overline{\unicode[STIX]{x1D714}_{i}}\unicode[STIX]{x1D61A}_{ij}^{\prime }\unicode[STIX]{x1D714}_{j}^{\prime }}$ (red line) is important only in the viscous sublayer, and it has a positive peak at $y^{+}\approx 3$ .

The second and third terms are more important for the generation mechanism of vortices in the buffer and log layers. The second term $\overline{\unicode[STIX]{x1D714}_{i}^{\prime }\overline{\unicode[STIX]{x1D61A}_{ij}}\unicode[STIX]{x1D714}_{j}^{\prime }}$ has a peak at $y^{+}\approx 10$ , and it decreases away from the wall. This trend means that the enstrophy produced by the strain-rate fields induced by the mean velocity is dominant in the buffer layer. As will be shown below, this corresponds to the event that the mean flow stretches the vortices in the buffer layer. However, this effect becomes less important in the log layer ( $y^{+}\gtrsim 10^{2}$ ), where the third term $\overline{\unicode[STIX]{x1D714}_{i}^{\prime }\unicode[STIX]{x1D61A}_{ij}^{\prime }\unicode[STIX]{x1D714}_{j}^{\prime }}$ is the largest. In other words, the enstrophy in the log layer, where a hierarchy of vortices exists, is generated predominantly by the stretching by the turbulent fluctuation rather than the mean flow.

Next, we investigate the scale dependence of the contribution of the rate-of-strain tensor to the enstrophy production. For this purpose, we introduce a quantity

(3.6) $$\begin{eqnarray}\displaystyle G_{S}(\unicode[STIX]{x1D70E}_{S}\rightarrow \unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})=\overline{\left.\left(\frac{\unicode[STIX]{x1D714}_{i}^{\prime (\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})}\unicode[STIX]{x1D61A}_{ij}^{\prime (\unicode[STIX]{x1D70E}_{S})}\unicode[STIX]{x1D714}_{j}^{\prime (\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})}}{|\unicode[STIX]{x1D714}_{i}^{\prime (\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})}|^{2}}\right)\right|}_{Q^{(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})}>0}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D714}_{i}^{\prime (\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})}$ is the fluctuation vorticity coarse-grained at the scale $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}$ , and $\unicode[STIX]{x1D61A}_{ij}^{\prime (\unicode[STIX]{x1D70E}_{S})}$ is the fluctuation rate-of-strain tensor coarse-grained at the scale $\unicode[STIX]{x1D70E}_{S}$ , both of which are calculated from the coarse-grained fluctuation velocity $\tilde{u} _{i}^{\prime }$ at $\unicode[STIX]{x1D70E}$ . Since $G_{S}(\unicode[STIX]{x1D70E}_{S}\rightarrow \unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})$ indicates the contribution of the strain rates coarse-grained at $\unicode[STIX]{x1D70E}_{S}$ to the stretching of the vortices coarse-grained at $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}$ , if $G_{S}(\unicode[STIX]{x1D70E}_{S}\rightarrow \unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})$ is positive (negative), it indicates the stretch (contraction) of the vortices. A similar quantity was defined by Goto et al. (Reference Goto, Saito and Kawahara2017) for turbulence in a periodic cube. Here, in order to evaluate the stretching of the vortices identified by the $Q$ criterion, we use the average conditioned by $Q^{(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})}>0$ . In addition, we also define contributions from the mean flow stretching as

(3.7) $$\begin{eqnarray}\displaystyle G_{M}(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})=\overline{\left.\left(\frac{\unicode[STIX]{x1D714}_{i}^{\prime (\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})}\overline{\unicode[STIX]{x1D61A}_{ij}}\unicode[STIX]{x1D714}_{j}^{\prime (\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})}}{|\unicode[STIX]{x1D714}_{i}^{\prime (\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})}|^{2}}\right)\right|}_{Q^{(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})}>0}. & & \displaystyle\end{eqnarray}$$

Figure 7.

Same as in figure 6 at (a) $y^{+}=160$ and (b) $y^{+}=60$ (i.e. $y/\unicode[STIX]{x1D6FF}_{99}\approx 0.3$ ) for lower Reynolds numbers (a) $Re_{\unicode[STIX]{x1D703}}=1640$ and (b) $440$ .

We evaluate these quantities defined by (3.6) and (3.7) at a streamwise location corresponding to $Re_{\unicode[STIX]{x1D703}}=3170$ . Results are plotted in figure 6 for three heights from the wall, (a) $y^{+}=300$ , (b) $100$ and (c) $40$ . Looking at the open symbols in figure 6(a), we can see that for smaller scales (i.e. $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}=10$ , $20$ and $40$ ) the contribution $G_{S}(2\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}\rightarrow \unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})$ from the strain-rate fields of twice-larger scales is the most significant. It is also seen that the contribution $G_{M}(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})$ from the mean shear (blue symbols) is less important than $G_{S}(2\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}\rightarrow \unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})$ . In contrast, for larger scales (i.e. $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}=80$ and $160$ ), denoted by closed symbols in figure 6(a), the contribution $G_{M}(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})$ from the mean shear is larger than $G_{S}(\unicode[STIX]{x1D70E}_{S}\rightarrow \unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})$ . These trends observed for the height $y^{+}=300$ (figure 6 a) are valid also for the height $y^{+}=100$ (figure 6 b). Therefore, an important conclusion drawn from figure 6(a,b) is that, in the log layer, the strain rates at the scale that is twice as large as the stretched vortices, rather than the mean flow, contribute most to the stretching of the small-scale vortices (say, $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}\lesssim y^{+}/5$ ; see figure 10 and the discussion of the figure). In the buffer layer, on the other hand, the mean-flow stretching is more important than the contribution from the strain-rate fields induced by the other scales. The evidence is given in figure 6(c), where the results for $y^{+}=40$ are shown. Among the contributions from the fluctuating strain-rate fields, the twice-larger scale contributes the most, but $G_{S}(2\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}\rightarrow \unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})$ is smaller than the contribution $G_{M}(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})$ from the mean shear. In summary, the mean flow contributes most to the stretching of large-scale vortices, i.e. the vortices whose size is of the order of the distance from the wall.

For comparison, we show in figure 7 results for relatively low Reynolds numbers $Re_{\unicode[STIX]{x1D703}}=1640$ (figure 7 a) and $Re_{\unicode[STIX]{x1D703}}=440$ (figure 7 b). Both results are for a height $y/\unicode[STIX]{x1D6FF}_{99}\approx 0.3$ . The result for $Re_{\unicode[STIX]{x1D703}}=440$ is obtained by another DNS (see appendix A). Similarly to the higher-Reynolds-number cases, among the contributions from fluctuating strain-rate fields, those twice as large as the stretched vortices contribute most significantly. However, the lower the Reynolds number is, the larger the contribution to the vortex stretching from the mean flow becomes.

We have shown that small-scale vortices are stretched predominantly either by the mean flow or by the strain-rate fields at the scale twice as large as themselves. The dominance depends on the distance from the wall, the size of the vortices and the Reynolds number. To quantitatively evaluate the dominance, we define the ratio

(3.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})=\frac{G_{S}(2\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}\rightarrow \unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})}{G_{M}(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})} & & \displaystyle\end{eqnarray}$$

of the contribution $G_{S}(2\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}\rightarrow \unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})$ of the coarse-grained strain rates to the stretching of the vortices coarse-grained at their half-scale and the contribution $G_{M}(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})$ of the mean flow. Here, we note that $\unicode[STIX]{x1D6E4}$ as well as $G_{S}$ and $G_{M}$ are functions not only of $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}$ but also of $y^{+}$ and $Re_{\unicode[STIX]{x1D703}}$ . For brevity, however, we omit the arguments $y^{+}$ and $Re_{\unicode[STIX]{x1D703}}$ of $\unicode[STIX]{x1D6E4}$ . When $\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})>1$ , the vortices at $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}$ are stretched predominantly by the strain rates twice as large as themselves; otherwise, they are stretched mainly by the mean flow.

Let us first investigate $\unicode[STIX]{x1D6E4}$ for the smallest-scale vortices. Figure 8 shows the results for $\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}=10)$ as a function of $Re_{\unicode[STIX]{x1D703}}$ and height $y^{+}$ . The ratio $\unicode[STIX]{x1D6E4}$ is less than $1$ for $y^{+}\lesssim 40$ for any Reynolds numbers, which means that the contribution to the stretching from the mean flow is larger for $y^{+}\lesssim 40$ , that is, even the very small vortices ( $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}=10$ ) in the buffer layer are stretched directly by the mean flow (see also figure 6 c). In contrast, in the log layer ( $y^{+}\gtrsim 100$ ), $\unicode[STIX]{x1D6E4}$ is greater than $1$ , which means that the contribution from the strain-rate fields induced by the twice-larger scale is more significant than from the mean shear.

Figure 8.

The ratio (3.8) between the contribution of strain rate coarse-grained at $\unicode[STIX]{x1D70E}_{S}^{+}=20$ to the stretching of vortices coarse-grained at $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}=10$ and the contribution of the mean flow as a function of $Re_{\unicode[STIX]{x1D703}}$ and the height $y^{+}$ . The blue line indicates the crossover ( $\unicode[STIX]{x1D6E4}=1$ ) of the dominance of the stretching from the mean flow to large-scale vortices. Only the region $y^{+}<\unicode[STIX]{x1D6FF}_{99}^{+}$ is plotted.

Figure 9.

Taylor-length-based Reynolds number $Re_{\unicode[STIX]{x1D706}}$ as a function of $Re_{\unicode[STIX]{x1D703}}$ and $y^{+}$ . Only the region $y^{+}<\unicode[STIX]{x1D6FF}_{99}^{+}$ is plotted.

To explain these observation, we plot the Taylor-length-based Reynolds number $Re_{\unicode[STIX]{x1D706}}=u_{rms}^{\prime }\unicode[STIX]{x1D706}/\unicode[STIX]{x1D708}$ as a function of $Re_{\unicode[STIX]{x1D703}}$ and $y^{+}$ in figure 9. Here, the Taylor length is defined by

(3.9) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}=\sqrt{\frac{{u_{rms}^{\prime }}^{2}}{\,\overline{(\unicode[STIX]{x2202}u^{\prime }/\unicode[STIX]{x2202}z)^{2}}\,}}. & & \displaystyle\end{eqnarray}$$

Comparing figures 8 and 9, we see that, in the buffer and log layers, $\unicode[STIX]{x1D6E4}$ is larger when $Re_{\unicode[STIX]{x1D706}}$ is larger. More precisely, in the buffer layer (say, $y^{+}\lesssim 40$ ), $Re_{\unicode[STIX]{x1D706}}$ is always smaller than $50$ , which means that there is no scale separation, and $\unicode[STIX]{x1D6E4}$ is always smaller than $1$ , which means that all vortices in the buffer layer are directly stretched by mean shear (figures 6 c, 7 b and 8). Figure 8 also shows that the crossover $\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}=10)\gtrless 1$ (the blue line in figure 8) occurs at $y^{+}$ between $40$ and $60$ . According to figure 9, this corresponds to $Re_{\unicode[STIX]{x1D706}}\approx 50$ . This value of $Re_{\unicode[STIX]{x1D706}}$ is reasonable, since it is known that the inertial range starts to appear for $Re_{\unicode[STIX]{x1D706}}\gtrsim 50$ . Furthermore, we can confirm that, for the locations with $Re_{\unicode[STIX]{x1D706}}\gtrsim 50$ (figure 9), $\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}=10)$ is always larger than $1$ (figure 8). Note that such points are located only in the log layer for $Re_{\unicode[STIX]{x1D703}}\gtrsim 1000$ . We emphasize that $\unicode[STIX]{x1D6E4}$ is always smaller than $1$ for any heights (except for the outer layer) when $Re_{\unicode[STIX]{x1D703}}\lesssim 500$ (figure 7 b).

Figure 10.

The ratio (3.8) between the contribution of strain rates coarse-grained at $2\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}$ to the stretching of vortices coarse-grained at $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}$ and the contribution of the mean flow as a function of $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}$ and $y^{+}$ for $Re_{\unicode[STIX]{x1D703}}=3170$ . Blue line, the crossover ( $\unicode[STIX]{x1D6E4}=1$ ) of the dominance of the stretching from the mean flow to large-scale vortices; red line, $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}=5\unicode[STIX]{x1D702}$ ; green dotted line, $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}=0.7L_{c}$ ; black dashed line, $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}=y/5$ . The contours are plotted by the interpolation of $\unicode[STIX]{x1D6E4}$ evaluated for the four cases ( $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}=10$ , $20$ , $40$ and $80$ ).

In figure 8, we have examined the contribution ratio $\unicode[STIX]{x1D6E4}$ (3.8) between the vortex stretching by twice-larger-scale vortices and the mean-flow stretching for the smallest scale ( $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}=10$ ) in the flow. However, although the scale ( $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}=10$ ) is smallest in the buffer layer, it is not necessarily smallest in the log layer. Therefore in figure 10 we show $\unicode[STIX]{x1D6E4}$ as a function of the filter scale $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}$ (namely, the size of vortices) and $y^{+}$ for a given Reynolds number ( $Re_{\unicode[STIX]{x1D703}}=3170$ ). The red line indicates $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}=5\unicode[STIX]{x1D702}$ as a function of $y^{+}$ . Since $5\unicode[STIX]{x1D702}$ is approximately the radius of the Kolmogorov-scale vortex tubes, we should look at $\unicode[STIX]{x1D6E4}$ in the region $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}\gtrsim 5\unicode[STIX]{x1D702}$ . For example, $\unicode[STIX]{x1D6E4}$ on the red line shows the dominance of the contribution to the stretching of the smallest vortices at each height. Looking at the value of $\unicode[STIX]{x1D6E4}$ on the red line, we see that the crossover ( $\unicode[STIX]{x1D6E4}\gtrless 1$ ) occurs at $y^{+}\approx 50$ , which is approximately the lower boundary of the log layer. It is also interesting to observe that the crossover (blue line) occurs at the height $y^{+}\approx 5\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}$ (black dashed line) for $y^{+}\lesssim 200$ (within the log layer), but this scaling does not hold in the outer layer ( $y^{+}\gtrsim 200$ for this $Re_{\unicode[STIX]{x1D703}}$ ). It is important to show that the height $y$ where $\unicode[STIX]{x1D6E4}=1$ is proportional to the Corrsin scale $L_{c}$ (Corrsin Reference Corrsin1958; Jiménez Reference Jiménez2013), which is defined by $L_{c}=\overline{\unicode[STIX]{x1D716}}^{1/2}S^{-3/2}$ with $S$ being the shear rate of the mean flow. In figure 10, the green dotted line indicates $0.7L_{c}$ , which is in good agreement with the blue line ( $\unicode[STIX]{x1D6E4}=1$ ). This observation is reasonable because the shearing time scale $S^{-1}$ and the cascade time scale (i.e. the eddy turnover time) $\ell ^{2/3}\hspace{2.22198pt}\overline{\unicode[STIX]{x1D716}}^{-1/3}$ are balanced at $\ell \approx L_{c}$ . Incidentally, Jiménez (Reference Jiménez2013) concluded that $L_{c}\approx 0.3y$ in the log layer, which is consistent with the above observation that the size $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}$ of vortices for $\unicode[STIX]{x1D6E4}=1$ is approximately $y/5$  ( ${\approx}(2/3)L_{c}\approx 0.7L_{c}$ ). In summary, figure 10 shows that $\unicode[STIX]{x1D6E4}<1$ in the buffer layer even for higher $Re_{\unicode[STIX]{x1D703}}$ (see also figures 6 c, 7 b and 8), whereas, in the log layer, $\unicode[STIX]{x1D6E4}>1$ in the scales between the red ( $5\unicode[STIX]{x1D702}$ ) and blue ( $y/5\approx 0.7L_{c}$ ) lines.

3.3 Alignment of vorticity and eigenvectors of rate-of-strain tensor

Figure 11.

Conditional p.d.f. of the alignment between the vorticity coarse-grained at (a) $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}=20$  ( $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}\approx 6\unicode[STIX]{x1D702}$ ) and (b) $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}=10$  ( $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}\approx 4\unicode[STIX]{x1D702}$ ) and the first eigenvector of the rate-of-strain tensor coarse-grained at $\unicode[STIX]{x1D70E}_{S}=1/2\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}$ (grey solid line), $\unicode[STIX]{x1D70E}_{S}=\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}$ (dashed line), $\unicode[STIX]{x1D70E}_{S}=2\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}$ (dot-dashed line), $\unicode[STIX]{x1D70E}_{S}=4\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}$ (double dot-dashed line, blue) and $\unicode[STIX]{x1D70E}_{S}=8\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}$ (triple dot-dashed line) at two different positions: (a) $y^{+}=300$ for $Re_{\unicode[STIX]{x1D703}}=3170$ ; and (b)  $y^{+}=60$ for $Re_{\unicode[STIX]{x1D703}}=440$ . The black solid line denotes the conditional p.d.f. of $|\!\cos \unicode[STIX]{x1D703}_{M}|$ .

Figure 12.

Conditional p.d.f. of the alignment between the vorticity coarse-grained at $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}=10$ (very light grey), $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}=20$ (light grey), $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}=40$ (grey), $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}=80$ (dark grey) and $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}=160$ (black) and the stretching direction of the mean flow at $y^{+}=300$ for $Re_{\unicode[STIX]{x1D703}}=3170$ .

We rewrite the production term of the fluctuation enstrophy as

(3.10) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D714}_{i}^{\prime }\unicode[STIX]{x1D61A}_{ij}^{\prime }\unicode[STIX]{x1D714}_{j}^{\prime }}{|\unicode[STIX]{x1D714}_{i}^{\prime }|^{2}}=s_{i}^{\prime }({\widehat{\boldsymbol{e}}_{i}}^{\prime }\boldsymbol{\cdot }\widehat{\unicode[STIX]{x1D74E}}^{\prime })^{2}, & & \displaystyle\end{eqnarray}$$

where $s_{i}^{\prime }$ ( $s_{1}^{\prime }\geqslant s_{2}^{\prime }\geqslant s_{3}^{\prime }$ ) are the eigenvalues of the fluctuation rate-of-strain tensor $\unicode[STIX]{x1D61A}_{ij}^{\prime }$ , ${\widehat{\boldsymbol{e}}_{i}}^{\prime }$ are the corresponding eigenvectors, and $\widehat{\unicode[STIX]{x1D74E}}^{\prime }$ is the normalized vorticity $\widehat{\unicode[STIX]{x1D74E}}^{\prime }=\unicode[STIX]{x1D74E}^{\prime }/\unicode[STIX]{x1D714}^{\prime }$ . Since $s_{1}^{\prime }+s_{2}^{\prime }+s_{3}^{\prime }=0$ for incompressible fluids, ${\widehat{\boldsymbol{e}}_{1}}^{\prime }$ corresponds to the direction with maximum stretching. In this section, we investigate the alignment between the vorticity and the stretching direction so that we can further clarify the picture of the vortex stretching in a turbulent boundary layer. For this purpose, we evaluate the probability density function (p.d.f.) of

(3.11) $$\begin{eqnarray}\displaystyle \cos \unicode[STIX]{x1D703}_{S}={\widehat{\boldsymbol{e}}_{1}}^{\,\prime (\unicode[STIX]{x1D70E}_{S})}\,\boldsymbol{\cdot }\,\widehat{\unicode[STIX]{x1D74E}}^{\prime (\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})}, & & \displaystyle\end{eqnarray}$$

where ${\widehat{\boldsymbol{e}}_{1}}^{\,\prime (\unicode[STIX]{x1D70E}_{S})}$ is the stretching direction of the fluctuation rate-of-strain tensor coarse-grained at scale $\unicode[STIX]{x1D70E}_{S}$ and $\widehat{\unicode[STIX]{x1D74E}}^{\prime (\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})}$ is the normalized fluctuation vorticity coarse-grained at scale $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}$ . We investigate the tendency of the alignment between the vortices at $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}$ and the stretching direction at $\unicode[STIX]{x1D70E}_{S}$ by calculating the p.d.f. of (3.11) using the conditional average for $Q^{(\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})}>0$ in the same way as in § 3.2. We also investigate the conditional p.d.f. of

(3.12) $$\begin{eqnarray}\displaystyle \cos \unicode[STIX]{x1D703}_{M}=\bar{\boldsymbol{e}}_{1}\,\boldsymbol{\cdot }\,\widehat{\unicode[STIX]{x1D74E}}^{\prime (\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})}, & & \displaystyle\end{eqnarray}$$

where $\bar{\boldsymbol{e}}_{1}$ denotes the stretching direction of the mean flow.

Fixing the filter scale $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}$ of the vorticity, we evaluate the conditional p.d.f. of $|\cos \unicode[STIX]{x1D703}_{S}|$ for various $\unicode[STIX]{x1D70E}_{S}$ for two different Reynolds numbers (figure 11 a, $Re_{\unicode[STIX]{x1D703}}=3170$ ; figure 11 b, $Re_{\unicode[STIX]{x1D703}}=440$ ). Figure 11(a) shows the result for the small scale ( $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}=20$ ) at $y^{+}=300$ (same as in figure 6 a) and figure 11(b) is for $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}=10$ at $y^{+}=60$ (same as in figure 7 b). In both cases, among different $\unicode[STIX]{x1D70E}_{S}$ , the tendency that ${\widehat{\boldsymbol{e}}_{1}}^{\,\prime (\unicode[STIX]{x1D70E}_{S})}\Vert \widehat{\unicode[STIX]{x1D74E}}^{\prime (\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}})}$ is the strongest for $\unicode[STIX]{x1D70E}_{S}^{+}=4\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D714}}^{+}$ (double dot-dashed blue lines in figure 11). Namely, the trend of parallel alignment between the vorticity at a small scale and the stretching direction at a scale four times as large is the strongest. This result is similar to the observation in decaying homogeneous isotropic turbulence that vortices at a given scale tend to align with the stretching direction at the scale $3$ – $5$ times as large as the given scale (Leung et al. Reference Leung, Swaminathan and Davidson2012). Despite this result, it is the strain-rate scale twice as large as the vortices that are dominant (figures 6 and 7) because the stretching rate is larger for smaller scales.

However, looking at the black solid curves in figure 11, we notice that the tendency that the coarse-grained fluctuation vorticity is parallel to the stretching direction of the mean flow is also relatively strong even for the higher-Reynolds-number case ( $Re_{\unicode[STIX]{x1D703}}=3170$ , figure 11 a). To investigate the scale dependence of the alignment between the coarse-grained fluctuation vorticity and the stretching direction of the mean flow, we plot the conditional p.d.f. of $|\cos \unicode[STIX]{x1D703}_{M}|$ for five different scales in figure 12. The Reynolds number and height are the same as in figure 11(a). Figure 12 shows that larger-scale vortices are more likely to align with the stretching direction of the mean flow. We confirm that this is the case also in the buffer layer (figures are omitted). This result is consistent with the fact that the contribution to the enstrophy production from the mean flow is larger for larger scales (figures 6 and 7). Note again that the enstrophy production depends not only on the alignment between the vorticity and the eigenvectors but also on the magnitude of the eigenvalues, and that the stretching rate is larger for smaller scales. Comparing the results for the two Reynolds numbers in figure 11, we also see that the parallel alignment of the vorticity with the stretching direction by the mean flow becomes more likely for lower Reynolds numbers. This is also consistent with the observation that the contribution to the stretching from the mean shear increases as the Reynolds number decreases (figure 7).

In summary, for larger scales at a given $Re_{\unicode[STIX]{x1D703}}$ , and for lower $Re_{\unicode[STIX]{x1D703}}$ at a given scale, vortices are more likely to align in the stretching direction of the mean shear. In other words, small-scale vortices become less aligned (i.e. more isotropic) as the Reynolds number increases, which is consistent with the visualization (figure 3). These are also consistent with the conclusions of the review by Jiménez (Reference Jiménez2013) that shear interacts with large-scale ‘eddies’, whereas small-scale vortices away from the wall decouple from the shear and become isotropic.